If a and b are two non-zero vectors such that ( a + b ) ⊥ a and (2 a + b ) ⊥ b, then prove that | b |=√ 2 | a |.

$$\begin{align*} & (\vec{\mathrm{a}}+\vec{\mathrm{b}}) \cdot \vec{\mathrm{a}}=0 \Rightarrow|\vec{\mathrm{a}}|^{2}+\vec{\mathrm{b}} \cdot \vec{\mathrm{a}}=0 \\ & (2 \vec{\mathrm{a}}+\vec{\mathrm{b}}) \cdot \vec{\mathrm{b}}=0 \Rightarrow 2 \vec{\mathrm{a}} \cdot \vec{\mathrm{b}}+|\vec{\mathrm{b}}|^{2}=0 \tag{2}\\ & 2\left(-|\vec{\mathrm{a}}|^{2}\right)+|\vec{\mathrm{b}}|^{2}=0 \text { \{Using (1) and (2)\}} \\ & |\vec{\mathrm{b}}|^{2}=2|\vec{\mathrm{a}}|^{2} \Rightarrow|\vec{\mathrm{b}}|=\sqrt{2}|\vec{\mathrm{a}}| \end{align*} $$